We use the standard definition of the boundary operations \(\partial _k\) . (See [1]}, Section 2.1.) Informally, the boundary of each \((k-1)\) -dimensional simplex that corresponds to a member \(A\in S_k\)  consists of all \((k-2)\) -dimensional simplices corresponding to \((k-1)\) -element subsets \(B\subset A\) . This definition naturally extends to any collection (“chain”) of \((k-1)\) -dimensional simplices that correspond to members of \(S_k\) , with multiplicities taken modulo 2.
